Chapter 13 · Introduction

Chapter 13 — Nuclei

Protons and neutrons are packed into a nucleus so small and so tightly bound that rearranging them releases about a million times more energy than any chemical reaction. This chapter measures nuclei, weighs them, and follows that energy from the mass defect to reactors and the Sun.

A nuclear power station the size of a few football pitches can supply electricity to a city of several million people, and it does it on a fuel delivery that would fit on the back of a lorry each year. A coal station of the same power burns a long freight train of coal every day. The Sun has been shining at almost the same brightness for four and a half billion years, and it will go on doing so for billions more. Hospitals use beams of protons, tuned to stop at exactly the depth of a tumour, and diagnostic tracers made in small accelerators. Behind all of these lies one tiny object: the atomic nucleus.

From the alpha-scattering experiments (Chapter 12 — Atoms) we know that nearly all of an atom's mass sits in a nucleus about $10^{-15}$ m across. In this chapter we open it up. We learn how atomic masses are measured and why they are quoted in a special unit, what a nucleus is made of, and how its size grows with the number of particles inside — which turns out to mean that every nucleus has the same, astonishing density. Then comes the key surprise: a nucleus weighs less than the protons and neutrons that make it. The missing mass, through Einstein's $E=mc^2$, is the energy that binds the nucleus together, and a graph of that binding energy per particle explains at a glance why splitting heavy nuclei (fission) and joining light ones (fusion) both release enormous amounts of energy. A short-range, very strong nuclear force holds the whole structure together against the electric repulsion of the protons.

Atomic masses (u)→Protons & neutrons→$R=R_0A^{1/3}$→$E=mc^2$→Mass defect & binding energy→Nuclear force→Fission & fusion

🎯 By the end of this chapter you can…

use the atomic mass unit, explain how a mass spectrometer separates isotopes, and find the average atomic mass of an element from isotopic abundances;
describe a nucleus by $Z$, $N$ and $A$, and classify nuclides as isotopes, isobars and isotones;
use $R=R_0A^{1/3}$ to find nuclear radii, and show that nuclear density is the same for all nuclei;
convert between mass and energy with $1\ \text{u}=931.5\ \text{MeV}/c^2$, and calculate the mass defect, binding energy and binding energy per nucleon of a nucleus;
describe the binding-energy-per-nucleon curve and the properties of the nuclear force, and use them to explain why fission and fusion release energy;
calculate the $Q$-value of a nuclear reaction, explain chain reactions, critical mass and the parts of a nuclear reactor, and describe fusion in the Sun;
solve board, JEE Main, JEE Advanced and NEET problems on all of the above.

Each bar is the energy released per kilogram of fuel, on a logarithmic scale (each grid step is ten times the last). It is computed from the energy of one reaction and the mass of the particles that take part. Chemical reactions rearrange electrons and release a few eV per event; nuclear reactions rearrange protons and neutrons and release millions of eV. The readouts show how much fuel a power station delivering $1000$ MW of heat would use in a year. Press Play to step through the fuels, or pick one. (Chemical values are typical, rounded figures.)

Fuel:

energy per event—

energy per kg—

fuel for 1000 MW, 1 year—

compared with coal—

Sections in this chapter

13.1

Atomic Masses, Composition and Size of the Nucleus

The atomic mass unit and mass spectrometry, protons and neutrons, isotopes, isobars and isotones, nuclear radius and nuclear density.

$R=R_0A^{1/3}$

Open section →13.2

Mass–Energy Equivalence, Binding Energy and the Nuclear Force

$E=mc^2$, the mass defect, binding energy and the binding-energy-per-nucleon curve, and the properties of the nuclear force.

$E_b=\Delta M\,c^2$

Open section →13.3

Nuclear Energy: Fission and Fusion

The $Q$-value of a reaction, fission of uranium, chain reactions and nuclear reactors, fusion and the energy of the Sun.

$Q=(m_\text{initial}-m_\text{final})\,c^2$

Open section →

✅ Before you start

You need Coulomb's law (Section 1.1), electrostatic potential energy and the electron volt, $1\ \text{eV}=1.6\times10^{-19}$ J (Section 2.1 and Section 2.2), and the circular path of a charged particle in a magnetic field, $r=mv/qB$ (Section 4.1), which is how a mass spectrometer sorts ions. From Chapter 12 — Atoms you need Rutherford's nuclear model of the atom and the distance of closest approach. From Class 11 Physics you need density, kinetic energy and the conservation of energy and momentum (Class 11 Physics, Chapter 5 — Work, Energy and Power). From mathematics you need powers and roots — especially cube roots — and powers of ten; no calculus is used in this chapter.

← 12.3 The Line Spectra of Hydrogen and de Broglie's Explanation 13.1 Atomic Masses, Composition and Size of the Nucleus →